% symbols
\newcommand{\ra}{\rightarrow}
\newcommand{\lra}{\longrightarrow}
\newcommand{\la}{\leftarrow}
\newcommand{\lla}{\longleftarrow}
\newcommand{\sqra}{\rightsquigarrow}
\newcommand{\sqlra}{\leftrightsquigarrow}
\newcommand{\inj}{\hookrightarrow}
\newcommand{\surj}{\twoheadrightarrow}
\newcommand{\sto}[1]{\stackrel{#1}{\longrightarrow}}
\newcommand{\lsto}[1]{\stackrel{#1}{\longleftarrow}}
\newcommand{\simto}{\sto{\sim}}
\newcommand{\lto}{\longmapsto}
\renewcommand{\vec}[1]{\overrightarrow{#1}}
\newcommand{\ilim}{\varinjlim\limits}
\newcommand{\plim}{\varprojlim\limits}
\newcommand{\wh}{\widehat}
\newcommand{\wt}{\widetilde}
\newcommand{\ov}{\overline}
\newcommand{\ul}{\underline}
\newcommand{\vare}{\varepsilon}
\newcommand{\varp}{\varphi}
\newcommand{\imply}{\Longrightarrow}
\newcommand{\rj}{\relbar\joinrel}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\mmid}{\parallel}
\newcommand{\pppi}{\frac{\partial\bar\partial}{\pi i}}
\font\cyr=wncyr10 \newcommand{\Sha}{\hbox{\cyr X}}


% functions
\newcommand{\fct}[4]{\begin{split}#1 &\lra #2 \\ #3 &\lto #4\end{split}}
\newcommand{\set}[1]{\left\{#1\right\}}
\newcommand{\pair}[1]{\langle{#1}\rangle}
\newcommand{\pmat}[4]{\begin{pmatrix}#1 & #2 \\ #3 & #4\end{pmatrix} }
\newcommand{\smat}[4]{\left(\begin{smallmatrix}#1 & #2 \\ #3 & #4\end{smallmatrix}\right) }
\newcommand{\bvec}[2]{\begin{bmatrix}#1 \\ #2\end{bmatrix} }
\newcommand{\norm}[1]{\!\parallel\!{#1}\!\parallel\!}
\newcommand{\env}[2]{\begin{#1}{#2}\end{#1}}
\newcommand{\envn}[3]{\begin{#1}[#2]{#3}\end{#1}}
\newcommand{\red}[1]{\textcolor{red}{#1}}
\newcommand{\hil}[3]{\left(\frac{{#1},{#2}}{#3}\right)}
\newcommand{\leg}[2]{\left(\frac{{#1}}{#2}\right)}


% categories
\newcommand{\cA}{{\mathsf{A}}}
\newcommand{\cB}{{\mathsf{B}}}
\newcommand{\cC}{{\mathsf{C}}}
\newcommand{\cAb}{{\mathsf{Ab}}}
\newcommand{\cBT}{{\mathsf{BT}}}
\newcommand{\cBun}{{\mathsf{Bun}}}
\newcommand{\cCharLoc}{{\mathsf{CharLoc}}}
\newcommand{\cCoh}{{\mathsf{Coh}}}
\newcommand{\cComm}{{\mathsf{Comm}}}
\newcommand{\cEt}{{\mathsf{Et}}}
\newcommand{\cFppf}{{\mathsf{Fppf}}}
\newcommand{\cFpqc}{{\mathsf{Fpqc}}}
\newcommand{\cFunc}{{\mathsf{Func}}}
\newcommand{\cGroups}{{\mathsf{Groups}}}
\newcommand{\cGrpd}{{\{\mathsf{Grpd}\}}}
\newcommand{\cHo}{{\mathsf{Ho}}}
\newcommand{\cIso}{{\mathsf{Iso}}}
\newcommand{\cLoc}{{\mathsf{Loc}}}
\newcommand{\cMod}{{\mathsf{Mod}}}
\newcommand{\cModFil}{{\mathsf{ModFil}}}
\newcommand{\cNilp}{{\mathsf{Nilp}}}
\newcommand{\cPerf}{{\mathsf{Perf}}}
\newcommand{\cPN}{{\mathsf{PN}}}
\newcommand{\cRep}{{\mathsf{Rep}}}
\newcommand{\cRings}{{\mathsf{Rings}}}
\newcommand{\cSets}{{\mathsf{Sets}}}
\newcommand{\cStack}{{\mathsf{Stack}}}
\newcommand{\cSch}{{\mathsf{Sch}}}
\newcommand{\cVect}{{\mathsf{Vect}}}
\newcommand{\cZar}{{\mathsf{Zar}}}
\newcommand{\cphimod}{{\varphi\txt{-}\mathsf{Mod}}}
\newcommand{\cphimodfil}{{\varphi\txt{-}\mathsf{ModFil}}}

% font
\newcommand{\rma}{{\mathrm{a}}}  \newcommand{\rmb}{{\mathrm{b}}}  \newcommand{\rmc}{{\mathrm{c}}}  \newcommand{\rmd}{{\mathrm{d}}}
\newcommand{\rme}{{\mathrm{e}}}  \newcommand{\rmf}{{\mathrm{f}}}  \newcommand{\rmg}{{\mathrm{g}}}  \newcommand{\rmh}{{\mathrm{h}}}
\newcommand{\rmi}{{\mathrm{i}}}  \newcommand{\rmj}{{\mathrm{j}}}  \newcommand{\rmk}{{\mathrm{k}}}  \newcommand{\rml}{{\mathrm{l}}}
\newcommand{\rmm}{{\mathrm{m}}}  \newcommand{\rmn}{{\mathrm{n}}}  \newcommand{\rmo}{{\mathrm{o}}}  \newcommand{\rmp}{{\mathrm{p}}}
\newcommand{\rmq}{{\mathrm{q}}}  \newcommand{\rmr}{{\mathrm{r}}}  \newcommand{\rms}{{\mathrm{s}}}  \newcommand{\rmt}{{\mathrm{t}}}
\newcommand{\rmu}{{\mathrm{u}}}  \newcommand{\rmv}{{\mathrm{v}}}  \newcommand{\rmw}{{\mathrm{w}}}  \newcommand{\rmx}{{\mathrm{x}}} \newcommand{\rmy}{{\mathrm{y}}}  \newcommand{\rmz}{{\mathrm{z}}}

\newcommand{\rmA}{{\mathrm{A}}}  \newcommand{\rmB}{{\mathrm{B}}}  \newcommand{\rmC}{{\mathrm{C}}}  \newcommand{\rmD}{{\mathrm{D}}}
\newcommand{\rmE}{{\mathrm{E}}}  \newcommand{\rmF}{{\mathrm{F}}}  \newcommand{\rmG}{{\mathrm{G}}}  \newcommand{\rmH}{{\mathrm{H}}}
\newcommand{\rmI}{{\mathrm{I}}}  \newcommand{\rmJ}{{\mathrm{J}}}  \newcommand{\rmK}{{\mathrm{K}}}  \newcommand{\rmL}{{\mathrm{L}}}
\newcommand{\rmM}{{\mathrm{M}}}  \newcommand{\rmN}{{\mathrm{N}}}  \newcommand{\rmO}{{\mathrm{O}}}  \newcommand{\rmP}{{\mathrm{P}}}
\newcommand{\rmQ}{{\mathrm{Q}}}  \newcommand{\rmR}{{\mathrm{R}}}  \newcommand{\rmS}{{\mathrm{S}}}  \newcommand{\rmT}{{\mathrm{T}}}
\newcommand{\rmU}{{\mathrm{U}}}  \newcommand{\rmV}{{\mathrm{V}}}  \newcommand{\rmW}{{\mathrm{W}}}  \newcommand{\rmX}{{\mathrm{X}}}
\newcommand{\rmY}{{\mathrm{Y}}}  \newcommand{\rmZ}{{\mathrm{Z}}}

\newcommand{\bfa}{{\mathbf{a}}}  \newcommand{\bfb}{{\mathbf{b}}}  \newcommand{\bfc}{{\mathbf{c}}}  \newcommand{\bfd}{{\mathbf{d}}}
\newcommand{\bfe}{{\mathbf{e}}}  \newcommand{\bff}{{\mathbf{f}}}  \newcommand{\bfg}{{\mathbf{g}}}  \newcommand{\bfh}{{\mathbf{h}}}
\newcommand{\bfi}{{\mathbf{i}}}  \newcommand{\bfj}{{\mathbf{j}}}  \newcommand{\bfk}{{\mathbf{k}}}  \newcommand{\bfl}{{\mathbf{l}}}
\newcommand{\bfm}{{\mathbf{m}}}  \newcommand{\bfn}{{\mathbf{n}}}  \newcommand{\bfo}{{\mathbf{o}}}  \newcommand{\bfp}{{\mathbf{p}}}
\newcommand{\bfq}{{\mathbf{q}}}  \newcommand{\bfr}{{\mathbf{r}}}  \newcommand{\bfs}{{\mathbf{s}}}  \newcommand{\bft}{{\mathbf{t}}}
\newcommand{\bfu}{{\mathbf{u}}}  \newcommand{\bfv}{{\mathbf{v}}}  \newcommand{\bfw}{{\mathbf{w}}}  \newcommand{\bfx}{{\mathbf{x}}} \newcommand{\bfy}{{\mathbf{y}}}  \newcommand{\bfz}{{\mathbf{z}}}

\newcommand{\bfA}{{\mathbf{A}}}  \newcommand{\bfB}{{\mathbf{B}}}  \newcommand{\bfC}{{\mathbf{C}}}  \newcommand{\bfD}{{\mathbf{D}}}
\newcommand{\bfE}{{\mathbf{E}}}  \newcommand{\bfF}{{\mathbf{F}}}  \newcommand{\bfG}{{\mathbf{G}}}  \newcommand{\bfH}{{\mathbf{H}}}
\newcommand{\bfI}{{\mathbf{I}}}  \newcommand{\bfJ}{{\mathbf{J}}}  \newcommand{\bfK}{{\mathbf{K}}}  \newcommand{\bfL}{{\mathbf{L}}}
\newcommand{\bfM}{{\mathbf{M}}}  \newcommand{\bfN}{{\mathbf{N}}}  \newcommand{\bfO}{{\mathbf{O}}}  \newcommand{\bfP}{{\mathbf{P}}}
\newcommand{\bfQ}{{\mathbf{Q}}}  \newcommand{\bfR}{{\mathbf{R}}}  \newcommand{\bfS}{{\mathbf{S}}}  \newcommand{\bfT}{{\mathbf{T}}}
\newcommand{\bfU}{{\mathbf{U}}}  \newcommand{\bfV}{{\mathbf{V}}}  \newcommand{\bfW}{{\mathbf{W}}}  \newcommand{\bfX}{{\mathbf{X}}}
\newcommand{\bfY}{{\mathbf{Y}}}  \newcommand{\bfZ}{{\mathbf{Z}}}

\newcommand{\BA}{{\mathbb{A}}}  \newcommand{\BB}{{\mathbb{B}}}  \newcommand{\BC}{{\mathbb{C}}}  \newcommand{\BD}{{\mathbb{D}}}  \newcommand{\BE}{{\mathbb{E}}}
\newcommand{\BF}{{\mathbb{F}}}  \newcommand{\BG}{{\mathbb{G}}}  \newcommand{\BH}{{\mathbb{H}}}  \newcommand{\BI}{{\mathbb{I}}}  \newcommand{\BJ}{{\mathbb{J}}}
\newcommand{\BK}{{\mathbb{K}}}  \newcommand{\BL}{{\mathbb{L}}}  \newcommand{\BM}{{\mathbb{M}}}  \newcommand{\BN}{{\mathbb{N}}}  \newcommand{\BO}{{\mathbb{O}}}
\newcommand{\BP}{{\mathbb{P}}}  \newcommand{\BQ}{{\mathbb{Q}}}  \newcommand{\BR}{{\mathbb{R}}}  \newcommand{\BS}{{\mathbb{S}}}  \newcommand{\BT}{{\mathbb{T}}}
\newcommand{\BU}{{\mathbb{U}}}  \newcommand{\BV}{{\mathbb{V}}}  \newcommand{\BW}{{\mathbb{W}}}  \newcommand{\BX}{{\mathbb{X}}}  \newcommand{\BY}{{\mathbb{Y}}}
\newcommand{\BZ}{{\mathbb{Z}}}

\newcommand{\CA}{{\mathcal{A}}} \newcommand{\CB}{{\mathcal{B}}} \newcommand{\CC}{{\mathcal{C}}}\renewcommand{\CD}{{\mathcal{D}}}\newcommand{\CE}{{\mathcal{E}}}
\newcommand{\CF}{{\mathcal{F}}} \newcommand{\CG}{{\mathcal{G}}} \newcommand{\CH}{{\mathcal{H}}} \newcommand{\CI}{{\mathcal{I}}} \newcommand{\CJ}{{\mathcal{J}}}
\newcommand{\CK}{{\mathcal{K}}} \newcommand{\CL}{{\mathcal{L}}} \newcommand{\CM}{{\mathcal{M}}} \newcommand{\CN}{{\mathcal{N}}} \newcommand{\CO}{{\mathcal{O}}}
\newcommand{\CP}{{\mathcal{P}}} \newcommand{\CQ}{{\mathcal{Q}}} \newcommand{\CR}{{\mathcal{R}}} \newcommand{\CS}{{\mathcal{S}}} \newcommand{\CT}{{\mathcal{T}}}
\newcommand{\CU}{{\mathcal{U}}} \newcommand{\CV}{{\mathcal{V}}} \newcommand{\CW}{{\mathcal{W}}} \newcommand{\CX}{{\mathcal{X}}} \newcommand{\CY}{{\mathcal{Y}}}
\newcommand{\CZ}{{\mathcal{Z}}}

\newcommand{\RA}{{\mathrm{A}}} \newcommand{\RB}{{\mathrm{B}}} \newcommand{\RC}{{\mathrm{C}}} \newcommand{\RD}{{\mathrm{D}}} \newcommand{\RE}{{\mathrm{E}}}
\newcommand{\RF}{{\mathrm{F}}} \newcommand{\RG}{{\mathrm{G}}} \newcommand{\RH}{{\mathrm{H}}} \newcommand{\RI}{{\mathrm{I}}} \newcommand{\RJ}{{\mathrm{J}}}
\newcommand{\RK}{{\mathrm{K}}} \newcommand{\RL}{{\mathrm{L}}} \newcommand{\RM}{{\mathrm{M}}} \newcommand{\RN}{{\mathrm{N}}} \newcommand{\RO}{{\mathrm{O}}}
\newcommand{\RP}{{\mathrm{P}}} \newcommand{\RQ}{{\mathrm{Q}}} \newcommand{\RR}{{\mathrm{R}}} \newcommand{\RS}{{\mathrm{S}}} \newcommand{\RT}{{\mathrm{T}}}
\newcommand{\RU}{{\mathrm{U}}} \newcommand{\RV}{{\mathrm{V}}} \newcommand{\RW}{{\mathrm{W}}} \newcommand{\RX}{{\mathrm{X}}} \newcommand{\RY}{{\mathrm{Y}}}
\newcommand{\RZ}{{\mathrm{Z}}}

\newcommand{\msa}{\mathscr{A}}  \newcommand{\msb}{\mathscr{B}}  \newcommand{\msc}{\mathscr{C}}  \newcommand{\msd}{\mathscr{D}}  \newcommand{\mse}{\mathscr{E}}
\newcommand{\msf}{\mathscr{F}}  \newcommand{\msg}{\mathscr{G}}  \newcommand{\msh}{\mathscr{H}}  \newcommand{\msi}{\mathscr{I}}  \newcommand{\msj}{\mathscr{J}}
\newcommand{\msk}{\mathscr{K}}  \newcommand{\msl}{\mathscr{L}}  \newcommand{\msm}{\mathscr{M}}  \newcommand{\msn}{\mathscr{N}}  \newcommand{\mso}{\mathscr{O}}
\newcommand{\msp}{\mathscr{P}}  \newcommand{\msq}{\mathscr{Q}}  \newcommand{\msr}{\mathscr{R}}  \newcommand{\mss}{\mathscr{S}}  \newcommand{\mst}{\mathscr{T}}
\newcommand{\msu}{\mathscr{U}}  \newcommand{\msv}{\mathscr{V}}  \newcommand{\msw}{\mathscr{W}}  \newcommand{\msx}{\mathscr{X}}  \newcommand{\msy}{\mathscr{Y}}
\newcommand{\msz}{\mathscr{Z}}

\newcommand{\fa}{{\mathfrak{a}}}\newcommand{\fb}{{\mathfrak{b}}}\newcommand{\fc}{{\mathfrak{c}}}\newcommand{\fd}{{\mathfrak{d}}}\newcommand{\fe}{{\mathfrak{e}}}
\newcommand{\ff}{{\mathfrak{f}}}\newcommand{\fg}{{\mathfrak{g}}}\newcommand{\fh}{{\mathfrak{h}}}\newcommand{\fii}{\mathfrak{i}} \newcommand{\fj}{{\mathfrak{j}}}
\newcommand{\fk}{{\mathfrak{k}}}\newcommand{\fl}{{\mathfrak{l}}}\newcommand{\fm}{{\mathfrak{m}}}\newcommand{\fn}{{\mathfrak{n}}}\newcommand{\fo}{{\mathfrak{o}}} \newcommand{\fp}{{\mathfrak{p}}}\newcommand{\fq}{{\mathfrak{q}}}\newcommand{\fr}{{\mathfrak{r}}}\newcommand{\fs}{{\mathfrak{s}}}\newcommand{\ft}{{\mathfrak{t}}}
\newcommand{\fu}{{\mathfrak{u}}}\newcommand{\fv}{{\mathfrak{v}}}\newcommand{\fw}{{\mathfrak{w}}}\newcommand{\fx}{{\mathfrak{x}}}\newcommand{\fy}{{\mathfrak{y}}}
\newcommand{\fz}{{\mathfrak{z}}}

\newcommand{\fA}{{\mathfrak{A}}}\newcommand{\fB}{{\mathfrak{B}}}\newcommand{\fC}{{\mathfrak{C}}}\newcommand{\fD}{{\mathfrak{D}}}\newcommand{\fE}{{\mathfrak{E}}}
\newcommand{\fF}{{\mathfrak{F}}}\newcommand{\fG}{{\mathfrak{G}}}\newcommand{\fH}{{\mathfrak{H}}}\newcommand{\fI}{{\mathfrak{I}}}\newcommand{\fJ}{{\mathfrak{J}}}
\newcommand{\fK}{{\mathfrak{K}}}\newcommand{\fL}{{\mathfrak{L}}}\newcommand{\fM}{{\mathfrak{M}}}\newcommand{\fN}{{\mathfrak{N}}}\newcommand{\fO}{{\mathfrak{O}}} \newcommand{\fP}{{\mathfrak{P}}}\newcommand{\fQ}{{\mathfrak{Q}}}\newcommand{\fR}{{\mathfrak{R}}}\newcommand{\fS}{{\mathfrak{S}}}\newcommand{\fT}{{\mathfrak{T}}}
\newcommand{\fU}{{\mathfrak{U}}}\newcommand{\fV}{{\mathfrak{V}}}\newcommand{\fW}{{\mathfrak{W}}} \newcommand{\fX}{{\mathfrak{X}}}\newcommand{\fY}{{\mathfrak{Y}}}\newcommand{\fZ}{{\mathfrak{Z}}}
% be careful about \fii

% a
\newcommand{\ab}{{\mathrm{ab}}}
\newcommand{\ad}{{\mathrm{ad}}}
\newcommand{\Ad}{{\mathrm{Ad}}}
\newcommand{\adele}{ad\'{e}le}
\newcommand{\Adele}{Ad\'{e}le}
\newcommand{\adeles}{ad\'{e}les}
\newcommand{\adelic}{ad\'{e}lic}
\newcommand{\AJ}{{\mathrm{AJ}}}
\newcommand{\alb}{{\mathrm{alb}}}
\newcommand{\Alb}{{\mathrm{Alb}}}
\newcommand{\alg}{{\mathrm{alg}}}
\newcommand{\an}{{\mathrm{an}}}
\newcommand{\Ann}{{\mathrm{Ann}}}
\newcommand{\Aut}{{\mathrm{Aut}}}
% b
\newcommand{\Bun}{{\mathrm{Bun}}}
\newcommand{\Br}{{\mathrm{Br}}}
\newcommand{\bs}{\backslash}
\newcommand{\BWt}{{\mathrm{BW}}}
% c
\newcommand{\can}{{\mathrm{can}}}
\newcommand{\cc}{{\mathrm{cc}}}
\newcommand{\cd}{{\mathrm{cd}}}
\newcommand{\ch}{{\mathrm{ch}}}
\newcommand{\Ch}{{\mathrm{Ch}}}
\newcommand{\Char}{{\mathrm{Char}}}
\newcommand{\Chow}{{\mathrm{CH}}}
\newcommand{\circB}{{\stackrel{\circ}{B}}}
\newcommand{\cl}{{\mathrm{cl}}}
\newcommand{\Cl}{{\mathrm{Cl}}}
\newcommand{\cm}{{\mathrm{cm}}}
\newcommand{\cod}{{\mathrm{cod}}}
\newcommand{\coker}{{\mathrm{coker}\,}}
\newcommand{\Coker}{{\mathrm{Coker}\,}}
\newcommand{\cond}{{\mathrm{cond}}}
\newcommand{\codim}{{\mathrm{codim}}}
\newcommand{\cont}{{\mathrm{cont}}}
\newcommand{\Conv}{{\mathrm{Conv}}}
\newcommand{\corr}{{\mathrm{corr}}}
\newcommand{\Corr}{{\mathrm{Corr}}}
\newcommand{\coim}{{\mathrm{coim}\,}}
\newcommand{\CoIm}{{\mathrm{CoIm}\,}}
\newcommand{\cris}{{\mathrm{cris}}}
\newcommand{\Cris}{{\mathrm{Cris}}}
\newcommand{\CRIS}{{\mathrm{CRIS}}}
\newcommand{\crit}{{\mathrm{crit}}} % critical
\newcommand{\crys}{{\mathrm{crys}}}
\newcommand{\cusp}{{\mathrm{cusp}}} % cuspidal
\newcommand{\CWt}{{\mathrm{CW}}} % Witt covector
\newcommand{\cyc}{{\mathrm{cyc}}} % cyclotomic
% d
\newcommand{\Def}{{\mathrm{Def}}} % deformation space
\newcommand{\diag}{{\mathrm{diag}}} % diagonal matrix
\newcommand{\diff}{\mathop{}\!\mathrm{d}} % differential
\newcommand{\disc}{{\mathrm{disc}}} % discriminant
\renewcommand{\div}{{\mathrm{div}}} % divisor
\newcommand{\Div}{{\mathrm{Div}}} % divisor group
\newcommand{\dR}{{\mathrm{dR}}} % de Rham
% e
\newcommand{\End}{{\mathrm{End}}} % endomorphism
\newcommand{\ess}{{\mathrm{ess}}} % essential
\newcommand{\et}{\text{\'{e}t}} % etale
\newcommand{\etale}{\'{e}tale} % etale
\newcommand{\Etale}{\'{E}tale} % etale
\newcommand{\Ext}{\mathrm{Ext}} % extension
\newcommand{\CExt}{\mathcal{E}\mathrm{xt}} % extension functor
% f
\newcommand{\Fil}{{\mathrm{Fil}}}
\newcommand{\Fix}{{\mathrm{Fix}}}
\newcommand{\fppf}{{\mathrm{fppf}}}
\newcommand{\Fr}{{\mathrm{Frac}}}
\newcommand{\Frob}{{\mathrm{Frob}}}
% g
\newcommand{\Ga}{\mathbb{G}_a}
\newcommand{\Gm}{\mathbb{G}_m}
\newcommand{\hGa}{\widehat{\mathbb{G}}_{a}}
\newcommand{\hGm}{\widehat{\mathbb{G}}_{m}}
\newcommand{\Gal}{{\mathrm{Gal}}}
\newcommand{\gl}{{\mathrm{gl}}}
\newcommand{\GL}{{\mathrm{GL}}}
\newcommand{\GO}{{\mathrm{GO}}}
\newcommand{\Gr}{{\mathrm{Gr}}}
\newcommand{\gr}{{\mathrm{gr}}}
\newcommand{\GSO}{{\mathrm{GSO}}}
\newcommand{\GSp}{{\mathrm{GSp}}}
\newcommand{\GSpin}{{\mathrm{GSpin}}}
\newcommand{\GU}{{\mathrm{GU}}}
% h
\newcommand{\hg}{{\mathrm{hg}}}
\newcommand{\Hk}{{\mathrm{Hk}}}
\newcommand{\HN}{{\mathrm{HN}}}
\newcommand{\Hom}{{\mathrm{Hom}}}
\newcommand{\CHom}{\mathcal{H}\mathrm{om}}
% i
\newcommand{\id}{{\mathrm{id}}}
\newcommand{\Id}{{\mathrm{Id}}}
\newcommand{\idele}{id\'{e}le}
\newcommand{\Idele}{Id\'{e}le}
\newcommand{\ideles}{id\'{e}les}
\renewcommand{\Im}{{\mathrm{Im}\,}}
\newcommand{\im}{{\mathrm{im}\,}}
\newcommand{\Ind}{{\mathrm{Ind}}}
\newcommand{\ind}{{\mathrm{ind}}}
\renewcommand{\inf}{{\mathrm{inf}}}
\newcommand{\Int}{{\mathrm{Int}}}
\newcommand{\inv}{{\mathrm{inv}}}
\newcommand{\Isom}{{\mathrm{Isom}}}
% j
\newcommand{\Jac}{{\mathrm{Jac}}}
\newcommand{\JL}{{\mathrm{JL}}}
% k
\newcommand{\Ker}{{\mathrm{Ker}\,}}
\newcommand{\KS}{{\mathrm{KS}}}
\newcommand{\Katz}{\mathrm{Katz}}
% l
\newcommand{\length}{\mathrm{length}}
\newcommand{\Li}{{\mathrm{Li}}}
\newcommand{\Lie}{{\mathrm{Lie}}}
\newcommand{\lt}{\mathrm{lt}}
\newcommand{\LT}{\mathcal{LT}}
% m
\newcommand{\MW}{{\mathrm{MW}}}
\renewcommand{\mod}{\, \mathrm{mod}\, }
\newcommand{\mom}{{\mathrm{mom}}}
\newcommand{\Mor}{{\mathrm{Mor}}}
\newcommand{\Morp}{{\mathrm{Morp}\,}}
% n
\newcommand{\new}{{\mathrm{new}}}
\newcommand{\Newt}{{\mathrm{Newt}}}
\newcommand{\NS}{{\mathrm{NS}}}
\newcommand{\ns}{{\mathrm{ns}}}
\newcommand{\Nm}{{\mathrm{Nm}}}
\newcommand{\Neron}{N\'{e}ron}
% o
\newcommand{\Obj}{{\mathrm{Obj}\,}}
\newcommand{\odd}{{\mathrm{odd}}}
\newcommand{\old}{{\mathrm{old}}}
\newcommand{\op}{{\mathrm{op}}}
\newcommand{\Orb}{{\mathrm{Orb}}}
\newcommand{\ord}{{\mathrm{ord}}}
% p
\newcommand{\pd}{{\mathrm{pd}}}
\newcommand{\Pet}{{\mathrm{Pet}}}
\newcommand{\PGL}{{\mathrm{PGL}}}
\newcommand{\Pic}{{\mathrm{Pic}}}
\newcommand{\Proj}{{\mathrm{Proj}}}
\newcommand{\proet}{\text{pro\'{e}t}}
\newcommand{\Poincare}{\text{Poincar\'{e}}}
\newcommand{\pr}{{\mathrm{pr}}}
\newcommand{\prim}{{\mathrm{prim}}}
% r
\newcommand{\Rad}{{\mathrm{Rad}}}
\newcommand{\rank}{{\mathrm{rank}}}
\renewcommand{\Re}{{\mathrm{Re}}}
\newcommand{\rec}{{\mathrm{rec}}}
\newcommand{\reg}{{\mathrm{reg}}}
\newcommand{\res}{{\mathrm{res}}}
\newcommand{\Res}{{\mathrm{Res}}}
\newcommand{\rig}{{\mathrm{rig}}}
\newcommand{\Rig}{{\mathrm{Rig}}}
\newcommand{\rk}{{\mathrm{rk}}}
\newcommand{\Ros}{{\mathrm{Ros}}}
\newcommand{\rs}{{\mathrm{rs}}}
% s
\newcommand{\sd}{{\mathrm{sd}}}
\newcommand{\Sel}{{\mathrm{Sel}}}
\newcommand{\sep}{{\mathrm{sep}}}
\newcommand{\sgn}{{\mathrm{sgn}}}
\newcommand{\Sh}{{\mathrm{Sh}}}
\newcommand{\Sht}{\mathrm{Sht}}
\newcommand{\Sim}{{\mathrm{Sim}}}
\newcommand{\sign}{{\mathrm{sign}}}
\newcommand{\SL}{{\mathrm{SL}}}
\newcommand{\SO}{{\mathrm{SO}}}
\newcommand{\Sp}{{\mathrm{Sp}}}
\newcommand{\Spa}{{\mathrm{Spa}}}
\newcommand{\Spec}{{\mathrm{Spec}}}
\newcommand{\Spf}{{\mathrm{Spf}}}
\newcommand{\Spin}{{\mathrm{Spin}}}
\newcommand{\Spm}{{\mathrm{Spm}}}
\newcommand{\srs}{{\mathrm{srs}}}
\newcommand{\rss}{{\mathrm{ss}}}
\newcommand{\st}{{\mathrm{st}}}
\newcommand{\Stab}{{\mathrm{Stab}}}
\newcommand{\SU}{{\mathrm{SU}}}
\newcommand{\Sym}{{\mathrm{Sym}}}
\newcommand{\sub}{{\mathrm{sub}}}
\newcommand{\supp}{{\mathrm{supp}}}
\newcommand{\Supp}{{\mathrm{Supp}}}
\newcommand{\suml}{\sum\limits}
\newcommand{\td}{{\mathrm{td}}}
\newcommand{\tor}{{\mathrm{tor}}}
\newcommand{\Tor}{{\mathrm{Tor}}}
\newcommand{\tors}{{\mathrm{tors}}}
\newcommand{\tr}{{\mathrm{tr}}}
\newcommand{\Tr}{{\mathrm{Tr}}}
\newcommand{\TSym}{{\mathrm{TSym}}}
\newcommand{\tw}{{\mathrm{tw}}}
% u
\newcommand{\uni}{{\mathrm{uni}}}
\newcommand{\univ}{\mathrm{univ}}
\newcommand{\ur}{{\mathrm{ur}}}
% v
\newcommand{\vQ}{{\breve \BQ}}
\newcommand{\vE}{{\breve E}}
\newcommand{\Ver}{{\mathrm{Ver}}}
\newcommand{\vF}{{\breve F}}
\newcommand{\vK}{{\breve K}}
\newcommand{\vol}{{\mathrm{vol}}}
\newcommand{\Vol}{{\mathrm{Vol}}}
% w
\newcommand{\wa}{{\mathrm{wa}}}

% theorems
\newtheorem{thm}{Theorem}[section]
\newtheorem{theo}[thm]{Theorem}
\newtheorem{theorem}[thm]{Theorem}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{coro}[thm]{Corollary}
\newtheorem{corollary}[thm]{Corollary}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{lemma}[thm]{Lemma}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{proposition}[thm]{Proposition}
\newtheorem{conj}[thm]{Conjecture}
\newtheorem{conjecture}[thm]{Conjecture}
\newtheorem{ques}[thm]{Question}
\newtheorem{question}[thm]{Question}
\newtheorem{openquestion}[thm]{Open Question}
\newtheorem{maintheorem}[thm]{Main Theorem}

\theoremstyle{definition}
\newtheorem{defn}[thm]{Definition}
\newtheorem{definition}[thm]{Definition}
\newtheorem{exam}[thm]{Example}
\newtheorem{example}[thm]{Example}
\newtheorem{exer}[thm]{Exercise}
\newtheorem{exercise}[thm]{Exercise}
\newtheorem{situ}[thm]{Situation}
\newtheorem{situation}[thm]{Situation}

\theoremstyle{remark}
\newtheorem{rem}[thm]{Remark}
\newtheorem{remark}[thm]{Remark}
\newtheorem{remarks}[thm]{Remarks}

\numberwithin{equation}{subsection}
